DeFranza

Linear Algebra
Introduction to Linear Algebra with Applications by Jim DeFranza and Daniel Gagliardi provides
the proper balance between computation, problem solving, and abstraction that will equip students with
the necessary skills and problem solving strategies to allow for a greater understanding and appreciation
of linear algebra and its numerous applications.

Abstract theory is essential to understanding how linear
algebra is applied.
Each concept is fully developed presenting natural connections
between topics giving students a working knowledge of the theory and techniques for each module
covered.
Applications have been carefully chosen to highlight the utility of linear algebra in
order to see the relevancy of the subject matter in other areas of science as well as in mathematics.

MD DALIM 976667 7/29/08 CYAN MAG YELO BLACK

Ranging from routine to more challenging, each exercise set extends the concepts
or techniques by asking the student to construct complete arguments. End of chapter True/False
questions help students connect concepts and facts presented in the chapter.
Examples are designed to develop intuition and prepare students to think more
conceptually about new topics as they are introduced.
Students are introduced to the study of linear algebra in a sequential and thorough
manner through an engaging writing style gaining a clear understanding of the theory essential for
applying linear algebra to mathematics or other fields of science.
Summaries conclude each section with important facts and techniques providing
students with easy access to the material needed to master the exercise sets.

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ISBN 978-0-07-353235-6
MHID 0-07-353235-5

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Linear Algebra

Introduction to Linear Algebra with Applications provides students with the necessary tools for success:

Linear Algebra


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INTRODUCTION TO
LINEAR ALGEBRA
WITH APPLICATIONS

Jim DeFranza
St. Lawrence University

Dan Gagliardi
SUNY Canton


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INTRODUCTION TO LINEAR ALGEBRA WITH APPLICATIONS
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New
York, NY 10020. Copyright  2009 by The McGraw-Hill Companies, Inc. All rights reserved. No part of this
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Some ancillaries, including electronic and print components, may not be available to customers outside the United States.
This book is printed on acid-free paper.
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ISBN 978–0–07–353235–6
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Library of Congress Cataloging-in-Publication Data

DeFranza, James, 1950–
Introduction to linear algebra / James DeFranza, Daniel Gagliardi. —1st ed.
p. cm.
Includes index.
ISBN 978–0–07–353235–6—ISBN 0–07–353235–5 (hard copy : alk. paper)
1. Algebras, Linear—Textbooks. 2. Algebras, Linear—Problems, exercises, etc. I. Gagliardi, Daniel. II. Title.
QA184.2.D44 2009
515 .5—dc22
2008026020

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To Regan, Sara, and David
—JD
To Robin, Zachary, Michael, and Eric
—DG


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About the Authors

Jim DeFranza was born in 1950 in Yonkers New York and grew up in Dobbs
Ferry New York on the Hudson River. Jim DeFranza is Professor of Mathematics
at St. Lawrence University in Canton New York where he has taught undergraduate
mathematics for 25 years. St. Lawrence University is a small Liberal Arts College
in upstate New York that prides itself in the close interaction that exists between
students and faculty. It is this many years of working closely with students that has
shaped this text in Linear Algebra and the other texts he has written. He received his
Ph.D. in Pure Mathematics from Kent State University in 1979. Dr. DeFranza has
coauthored PRECALCULUS, Fourth Edition and two other texts in single variable
and multivariable calculus. Dr. DeFranza has also published a dozen research articles
in the areas of Sequence Spaces and Classical Summability Theory. Jim is married
and has two children David and Sara. Jim and his wife Regan live outside of Canton
New York in a 150 year old farm house.
Daniel Gagliardi is an Assistant Professor of Mathematics at SUNY Canton, in
Canton New York. Dr. Gagliardi began his career as a software engineer at IBM
in East Fishkill New York writing programs to support semiconductor development
and manufacturing. He received his Ph.D. in Pure Mathematics from North Carolina
State University in 2003 under the supervision of Aloysius Helminck. Dr. Gagliardi’s
principle area of research is in Symmetric Spaces. In particular, his current work
is concerned with developing algorithmic formulations to describe the fine structure
(characters and Weyl groups) of local symmetric spaces. Dr. Gagliardi also does
research in Graph Theory. His focus there is on the graphical realization of certain
types of sequences. In addition to his work as a mathematician, Dr. Gagliardi is an
accomplished double bassist and has recently recorded a CD of jazz standards with
Author/Pianist Bill Vitek. Dr. Gagliardi lives in northern New York in the picturesque
Saint Lawrence River Valley with his wife Robin, and children Zachary, Michael,

and Eric.

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Contents

Preface ix
CHAPTER

1

Systems of Linear Equations and Matrices

1

1.1 Systems of Linear Equations 2
Exercise Set 1.1 12
1.2 Matrices and Elementary Row Operations 14
Exercise Set 1.2 23
1.3 Matrix Algebra 26
Exercise Set 1.3 37
1.4 The Inverse of a Square Matrix 39
Exercise Set 1.4 45
1.5 Matrix Equations 48
Exercise Set 1.5 51

1.6 Determinants 54
Exercise Set 1.6 65
1.7 Elementary Matrices and LU Factorization 68
Exercise Set 1.7 77
1.8 Applications of Systems of Linear Equations 79
Exercise Set 1.8 84
Review Exercises 89
Chapter Test 90
CHAPTER

2

Linear Combinations and Linear Independence
2.1 Vectors in ‫ޒ‬n 94
Exercise Set 2.1 99
2.2 Linear Combinations 101
Exercise Set 2.2 108
2.3 Linear Independence 111
Exercise Set 2.3 120
Review Exercises 123
Chapter Test 125

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Contents

CHAPTER

3

Vector Spaces

127

3.1 Definition of a Vector Space 129
Exercise Set 3.1 137
3.2 Subspaces 140
Exercise Set 3.2 154
3.3 Basis and Dimension 156
Exercise Set 3.3 171
3.4 Coordinates and Change of Basis 173
Exercise Set 3.4 182
3.5 Application: Differential Equations 185
Exercise Set 3.5 193
Review Exercises 194
Chapter Test 195
CHAPTER

4

Linear Transformations

199


4.1 Linear Transformations 200
Exercise Set 4.1 211
4.2 The Null Space and Range 214
Exercise Set 4.2 223
4.3 Isomorphisms 226
Exercise Set 4.3 233
4.4 Matrix Representation of a Linear Transformation
Exercise Set 4.4 245
4.5 Similarity 249
Exercise Set 4.5 253
4.6 Application: Computer Graphics 255
Exercise Set 4.6 268
Review Exercises 270
Chapter Test 272
CHAPTER

5

235

Eigenvalues and Eigenvectors 275
5.1 Eigenvalues and Eigenvectors 276
Exercise Set 5.1 285
5.2 Diagonalization 287
Exercise Set 5.2 298
5.3 Application: Systems of Linear Differential Equations
Exercise Set 5.3 309

300


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Contents

5.4 Application: Markov Chains
Exercise Set 5.4 315
Review Exercises 316
Chapter Test 318
CHAPTER

6

Inner Product Spaces

310

321

6.1 The Dot Product on ‫ޒ‬n 323
Exercise Set 6.1 331
6.2 Inner Product Spaces 333
Exercise Set 6.2 341
6.3 Orthonormal Bases 342
Exercise Set 6.3 352

6.4 Orthogonal Complements 355
Exercise Set 6.4 364
6.5 Application: Least Squares Approximation 366
Exercise Set 6.5 375
6.6 Diagonalization of Symmetric Matrices 377
Exercise Set 6.6 383
6.7 Application: Quadratic Forms 385
Exercise Set 6.7 392
6.8 Application: Singular Value Decomposition 392
Exercise Set 6.8 403
Review Exercises 404
Chapter Test 406

Appendix 409
Answers to Odd-Numbered Exercises 440
Index 479


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Preface

Introduction to Linear Algebra with Applications is an introductory text targeted to
second-year or advanced first-year undergraduate students. The organization of this
text is motivated by what our experience tells us are the essential concepts that students
should master in a one-semester undergraduate linear algebra course. The centerpiece
of our philosophy regarding the presentation of the material is that each topic should
be fully developed before the reader moves onto the next. In addition, there should be

a natural connection between topics. We take great care to meet both of these objectives. This allows us to stay on task so that each topic can be covered with the depth
required before progression to the next logical one. As a result, the reader is prepared
for each new unit, and there is no need to repeat a concept in a subsequent chapter
when it is utilized.
Linear algebra is taken early in an undergraduate curriculum and yet offers the
opportunity to introduce the importance of abstraction, not only in mathematics, but in
many other areas where linear algebra is used. Our approach is to take advantage of this
opportunity by presenting abstract vector spaces as early as possible. Throughout the
text, we are mindful of the difficulties that students at this level have with abstraction
and introduce new concepts first through examples which gently illustrate the idea.
To motivate the definition of an abstract vector space, and the subtle concept of
linear independence, we use addition and scalar multiplication of vectors in Euclidean
space. We have strived to create a balance among computation, problem solving, and
abstraction. This approach equips students with the necessary skills and problemsolving strategies in an abstract setting that allows for a greater understanding and
appreciation for the numerous applications of the subject.

Pedagogical Features
1. Linear systems, matrix algebra, and determinants: We have given a streamlined, but complete, discussion of solving linear systems, matrix algebra, determinants, and their connection in Chap. 1. Computational techniques are introduced,
and a number of theorems are proved. In this way, students can hone their
problem-solving skills while beginning to develop a conceptual sense of the fundamental ideas of linear algebra. Determinants are no longer central in linear
algebra, and we believe that in a course at this level, only a few lectures should
be devoted to the topic. For this reason we have presented all the essentials on
determinants, including their connection to linear systems and matrix inverses,
in Chap. 1. This choice also enables us to use determinants as a theoretical tool
throughout the text whenever the need arises.

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2. Vectors: Vectors are introduced in Chap. 1, providing students with a familiar
structure to work with as they start to explore the properties which are used later
to characterize abstract vector spaces.
3. Linear independence: We have found that many students have difficulties with
linear combinations and the concept of linear independence. These ideas are fundamental to linear algebra and are essential to almost every topic after linear
systems. When students fail to grasp them, the full benefits of the course cannot
be realized. In Introduction to Linear Algebra with Applications we have devoted
Chap. 2 to a careful exposition of linear combinations and linear independence
in the context of Euclidean space. This serves several purposes. First, by placing
these concepts in a separate chapter their importance in linear algebra is highlighted. Second, an instructor using the text can give exclusive focus to these ideas
before applying them to other problems and situations. Third, many of the important ramifications of linear combinations and linear independence are considered
in the familiar territory of Euclidean spaces.
4. Euclidean spaces ‫ޒ‬n : The Euclidean spaces and their algebraic properties are
introduced in Chap. 2 and are used as a model for the abstract vectors spaces of
Chap. 3. We have found that this approach works well for students with limited
exposure to abstraction at this level.
5. Geometric representations: Whenever possible, we include figures with geometric representations and interpretations to illuminate the ideas being presented.
6. New concepts: New concepts are almost always introduced first through concrete
examples. Formal definitions and theorems are then given to describe the situation
in general. Additional examples are also provided to further develop the new idea
and to explore it in greater depth.
7. True/false chapter tests: Each chapter ends with a true/false Chapter Test with
approximately 40 questions. These questions are designed to help the student
connect concepts and better understand the facts presented in the chapter.

8. Rigor and intuition: The approach we have taken attempts to strike a balance
between presenting a rigorous development of linear algebra and building intuition. For example, we have chosen to omit the proofs for theorems that are not
especially enlightening or that contain excessive computations. When a proof is
not present, we include a motivating discussion describing the importance and
use of the result and, if possible, the idea behind a proof.
9. Abstract vector spaces: We have positioned abstract vector spaces as a central
topic within Introduction to Linear Algebra with Applications by placing their
introduction as early as possible in Chap. 3. We do this to ensure that abstract
vector spaces receive the appropriate emphasis. In a typical undergraduate mathematics curriculum, a course on linear algebra is the first time that students are
exposed to this level of abstraction. However, Euclidean spaces still play a central
role in our approach because of their familiarity and since they are so widely
used. At the end of this chapter, we include a section on differential equations
which underscores the need for the abstract theory of vector spaces.


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10. Section fact summaries: Each section ends with a summary of the important facts
and techniques established in the section. They are written, whenever possible,
using nontechnical language and mostly without notation. These summaries are
not meant to give a recapitulation of the details and formulas of the section;
rather they are designed to give an overview of the main ideas of the section.
Our intention is to help students to make connections between the concepts of
the section as they survey the topic from a greater vantage point.


Applications
Over the last few decades the applications of linear algebra have mushroomed, increasing not only in their numbers, but also in the diversity of fields to which they apply.
Much of this growth is fueled by the power of modern computers and the availability
of computer algebra systems used to carry out computations for problems involving
large matrices. This impressive power has made linear algebra more relevant than
ever. Recently, a consortium of mathematics educators has placed its importance, relative to applications, second only to calculus. Increasingly, universities are offering
courses in linear algebra that are specifically geared toward its applications. Whether
the intended audience is engineering, economics, science, or mathematics students,
the abstract theory is essential to understanding how linear algebra is applied.
In this text our introduction to the applications of linear algebra begins in Sec. 1.8
where we show how linear systems can be used to solve problems related to chemistry,
engineering, economics, nutrition, and urban planning. However, many types of applications involve the more sophisticated concepts we develop in the text. These applications require the theoretical notions beyond the basic ideas of Chap. 1, and are
presented at the end of a chapter as soon as the required background material is completed. Naturally, we have had to limit the number of applications considered. It is our
hope that the topics we have chosen will interest the reader and lead to further inquiry.
Specifically, in Sec. 4.6, we discuss the role of linear algebra in computer graphics. An introduction to the connection between differential equations and linear algebra
is given in Secs. 3.5 and 5.3. Markov chains and quadratic forms are examined in
Secs. 5.4 and 6.7, respectively. Section 6.5 focuses on the problem of finding approximate solutions to inconsistent linear systems. One of the most familiar applications
here is the problem of finding the equation of a line that best fits a set of data points.
Finally, in Sec. 6.8 we consider the singular value decomposition of a matrix and its
application to data compression.

Technology
Computations are an integral part of any introductory course in mathematics and
certainly in linear algebra. To gain mastery of the techniques, we encourage the student
to solve as many problems as possible by hand. That said, we also encourage the
student to make appropriate use of the available technologies designed to facilitate,
or to completely carry out, some of the more tedious computations. For example, it
is quite reasonable to use a computer algebra system, such as MAPLE or MATLAB,



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Preface

to row-reduce a large matrix. Our approach in Introduction to Linear Algebra with
Applications is to assume that some form of technology will be used, but leave the
choice to the individual instructor and student. We do not think that it is necessary to
include discussions or exercises that use particular software. Note that this text can be
used with or without technology. The degree to which it is used is left to the discretion
of the instructor. From our own experience, we have found that Scientific Notebook,TM
which offers a front end for LATEX along with menu access to the computer algebra
system MuPad, allows the student to gain experience using technology to carry out
computations while learning to write clear mathematics. Another option is to use LATEX
for writing mathematics and a computer algebra system to perform computations.
Another aspect of technology in linear algebra has to do with the accuracy and
efficiency of computations. Some applications, such as those related to Internet search
engines, involve very large matrices which require extensive processing. Moreover, the
accuracy of the results can be affected by computer roundoff error. For example, using
the characteristic equation to find the eigenvalues of a large matrix is not feasible.
Overcoming problems of this kind is extremely important. The field of study known as
numerical linear algebra is an area of vibrant research for both software engineers and
applied mathematicians who are concerned with developing practical solutions. In our
text, the fundamental concepts of linear algebra are introduced using simple examples.
However, students should be made aware of the computational difficulties that arise
when extending these ideas beyond the small matrices used in the illustrations.

Other Features

1. Chapter openers: The opening remarks for each chapter describe an application
that is directly related to the material in the chapter. These provide additional
motivation and emphasize the relevance of the material that is about to be covered.
2. Writing style: The writing style is clear, engaging, and easy to follow. Important new concepts are first introduced with examples to help develop the reader’s
intuition. We limit the use of jargon and provide explanations that are as readerfriendly as possible. Every explanation is crafted with the student in mind. Introduction to Linear Algebra with Applications is specifically designed to be a
readable text from which a student can learn the fundamental concepts in linear
algebra.
3. Exercise sets: Exercise sets are organized with routine exercises at the beginning
and the more difficult problems toward the end. There is a mix of computational
and theoretical exercises with some requiring proof. The early portion of each
exercise set tests the student’s ability to apply the basic concepts. These exercises
are primarily computational, and their solutions follow from the worked examples
in the section. The latter portion of each exercise set extends the concepts and
techniques by asking the student to construct complete arguments.
4. Review exercise sets: The review exercise sets are organized as sample exams
with 10 exercises. These exercises tend to have multiple parts, which connect
the various techniques and concepts presented in the text. At least one problem
in each of these sets presents a new idea in the context of the material of the
chapter.


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Preface

xiii

5. Length: The length of the text reflects the fact that it is specifically designed for
a one-semester course in linear algebra at the undergraduate level.

6. Appendix: The appendix contains background material on the algebra of sets,
functions, techniques of proof, and mathematical induction. With this feature, the
instructor is able to cover, as needed, topics that are typically included in a Bridge
Course to higher mathematics.

Course Outline
The topics we have chosen for Introduction to Linear Algebra with Applications
closely follow those commonly covered in a first introductory course. The order in
which we present these topics reflects our approach and preferences for emphasis.
Nevertheless, we have written the text to be flexible, allowing for some permutations
of the order of topics without any loss of consistency. In Chap. 1 we present all the
basic material on linear systems, matrix algebra, determinants, elementary matrices,
and the LU decomposition. Chap. 2 is entirely devoted to a careful exposition of linear combinations and linear independence in ‫ޒ‬n . We have found that many students
have difficulty with these essential concepts. The addition of this chapter gives us
the opportunity to develop all the important ideas in a familiar setting. As mentioned
earlier, to emphasize the importance of abstract vector spaces, we have positioned
their introduction as early as possible in Chap. 3. Also, in Chap. 3 is a discussion
of subspaces, bases, and coordinates. Linear transformations between vector spaces
are the subject of Chap. 4. We give descriptions of the null space and range of a
linear transformation at the beginning of the chapter, and later we show that every
finite dimensional vector space, of dimension n, is isomorphic to ‫ޒ‬n . Also, in Chap. 4
we introduce the four fundamental subspaces of a matrix and discuss the action of an
m × n matrix on a vector in ‫ޒ‬n . Chap. 5 is concerned with eigenvalues and eigenvectors. An abundance of examples are given to illustrate the techniques of computing
eigenvalues and finding the corresponding eigenvectors. We discuss the algebraic and
geometric multiplicities of eigenvalues and give criteria for when a square matrix is
diagonalizable. In Chap. 6, using ‫ޒ‬n as a model, we show how a geometry can be
defined on a vector space by means of an inner product. We also give a description
of the Gram-Schmidt process used to find an orthonormal basis for an inner product
space and present material on orthogonal complements. At the end of this chapter we
discuss the singular value decomposition of an m × n matrix. The Appendix contains

a brief summary of some topics found in a Bridge Course to higher mathematics.
Here we include material on the algebra of sets, functions, techniques of proof, and
mathematical induction. Application sections are placed at the end of chapters as soon
as the requisite background material has been covered.

Supplements
1. Instructor solutions manual: This manual contains detailed solutions to all
exercises.
2. Student solutions manual: This manual contains detailed solutions to oddnumbered exercises.


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3. Text website www.mhhe.com/defranza: This website accompanies the text and
is available for both students and their instructors. Students will be able to access
self-assessment quizzes and extra examples for each section and end of chapter
cumulative quizzes. In addition to these assets, instructors will be able to access
additional quizzes, sample exams, the end of chapter true/false tests, and the
Instructor’s Solutions Manual.

Acknowledgments
We would like to give our heartfelt thanks to the many individuals who reviewed
the manuscript at various stages of its development. Their thoughtful comments and
excellent suggestions have helped us enormously with our efforts to realize our vision
of a reader-friendly introductory text on linear algebra.
We would also like to give special thanks to David Meel of Bowling Green
State University, Bowling Green, Ohio, for his thorough review of the manuscript and

insightful comments that have improved the exposition of the material in the text.
We are also grateful to Ernie Stitzinger of North Carolina State University who had
the tiring task of checking the complete manuscript for accuracy, including all the
exercises. A very special thanks goes to our editors (and facilitators), Liz Covello
(Sr. Sponsoring Editor), Michelle Driscoll (Developmental Editor), and Joyce Watters
(Project Manager) who have helped us in more ways than we can name, from the
inception of this project to its completion. On a personal level, we would like to
thank our wives, Regan DeFranza and Robin Gagliardi, for their love and support;
and our students at Saint Lawrence University and SUNY Canton who provided the
motivation to write the text. Finally, we want to express our gratitude to the staff
at McGraw-Hill Higher Education, Inc., for their work in taking our manuscript and
producing the text.


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List of Reviewers
Marie Aratari, Oakland Community College
Cik Azizah, Universiti Utara Malaysia (UUM)
Przcmyslaw Bogacki, Old Dominion University
Rita Chattopadhyay, Eastern Michigan University
Eugene Don, Queens College
Lou Giannini, Curtin University of Technology
Gregory Gibson, North Carolina A&T University
Mark Gockenback, Michigan Technological University

Dr. Leong Wah June, Universiti Putra Malaysia
Cerry Klein, University of Missouri–Columbia
Kevin Knudson, Mississippi State University
Hyungiun Ko, Yonsei University
Jacob Kogan, University of Maryland–Baltimore County
David Meel, Bowling Green State University
Martin Nakashima, California State Poly University–Pomona
Eugene Spiegel, University of Connecticut–Storrs
Dr. Hajar Sulaiman, Universiti Sains Malaysia (USM)
Gnana Bhaskar Tenali, Florida Institute of Technology–Melbourne
Peter Wolfe, University of Maryland–College Park

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To The Student

You are probably taking this course early in your undergraduate studies after two or
three semesters of calculus, and most likely in your second year. Like calculus, linear
algebra is a subject with elegant theory and many diverse applications. However,
in this course you will be exposed to abstraction at a much higher level. To help
with this transition, some colleges and universities offer a Bridge Course to Higher
Mathematics. If you have not already taken such a course, this may likely be the
first mathematics course where you will be expected to read and understand proofs of
theorems, provide proofs of results as part of the exercise sets, and apply the concepts
presented. All this is in the context of a specific body of knowledge. If you approach

this task with an open mind and a willingness to read the text, some parts perhaps
more than once, it will be an exciting and rewarding experience. Whether you are
taking this course as part of a mathematics major or because linear algebra is applied
in your specific area of study, a clear understanding of the theory is essential for
applying the concepts of linear algebra to mathematics or other fields of science. The
solved examples and exercises in the text are designed to prepare you for the types
of problems you can expect to see in this course and other more advanced courses in
mathematics. The organization of the material is based on our philosophy that each
topic should be fully developed before readers move onto the next. The image of a tree
on the front cover of the text is a metaphor for this learning strategy. It is particularly
applicable to the study of mathematics. The trunk of the tree represents the material
that forms the basis for everything that comes afterward. In our text, this material is
contained in Chaps. 1 through 4. All other branches of the tree, representing more
advanced topics and applications, extend from the foundational material of the trunk or
from the ancillary material of the intervening branches. We have specifically designed
our text so that you can read it and learn the concepts of linear algebra in a sequential
and thorough manner. If you remain committed to learning this beautiful subject, the
rewards will be significant in other courses you may take, and in your professional
career. Good luck!
Jim DeFranza

Dan Gagliardi


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Applications Index

Aircraft Design, 199
Astronomy, 61
Average Global Temperatures, 371
Balancing Chemical Equations, 1, 79, 84
Carbon Emissions, 321
Cellular Phone Subscribers, 86
Computer Graphics, 199, 255
Cumulative HIV Infections, 376
Data Compression, 401
Differential Equations, 185
Digital Transmissions, 127, 402
Economics, 82, 86, 87
Electrical Networks, 88, 191
Engineering, 185, 191
Epidemics, 87, 315
Exponential Growth and Decay, 186
Fitting Data with a Curve, 11, 61, 366, 372, 376
Fourier Approximation, 373
Infant Mortality Rates, 376
Least Squares Approximation, 321, 366
Linear Regression, 371
Markov Chains, 275, 310
Mass Transit Systems, 315
Mixing, 307, 310
National Health Care, 86
Networks, 79, 85, 89
Nutrition, 81, 85

Photosynthesis, 1
Plant Breading, 315
Population Distribution, 87, 275, 315
Powers of a Matrix, 287
Predator–Prey Models, 300
Projections, 265, 343, 345, 361, 365
Quadratic Forms, 385
Rotation of Axes, 385
Search Engines, 276

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Applications Index

Signal Processing, 93
Singular Value Decomposition, 392
Systems of Differential Equations, 300
Thermal Equilibrium, 88, 310
Vibrating Spring, 192, 194
World Hydroelectricity Use, 376
World Population, 376


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CHAPTER

Systems of Linear
Equations and Matrices
CHAPTER OUTLINE

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8

Systems of Linear Equations 2
Matrices and Elementary Row Operations 14
Matrix Algebra 26
The Inverse of a Square Matrix 39
Matrix Equations 48
Determinants 54
Elementary Matrices and LU Factorization 68
Applications of Systems of Linear Equations 79

n the process of photosynthesis solar energy
is converted into forms that are used by living
organisms. The chemical reaction that occurs in

the leaves of plants converts carbon dioxide and
water to carbohydrates with the release of oxygen.
The chemical equation of the reaction takes the
form

I

aCO2 + bH2 O → cO2 + dC6 H12 O6
where a, b, c, and d are some positive whole
numbers. The law of conservation of mass states
that the total mass of all substances present before
and after a chemical reaction remains the same.
That is, atoms are neither created nor destroyed Photograph by Jan Smith/RF
in a chemical reaction, so chemical equations must be balanced. To balance the photosynthesis reaction equation, the same number of carbon atoms must appear on both
sides of the equation, so
a = 6d
The same number of oxygen atoms must appear on both sides, so
2a + b = 2c + 6d
and the same number of hydrogen atoms must appear on both sides, so
2b = 12d
1


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Chapter 1 Systems of Linear Equations and Matrices

This gives us the system of three linear equations in four variables



− 6d = 0
 a
2a + b − 2c − 6d = 0


2b
− 12d = 0
Any positive integers a, b, c, and d that satisfy all three equations are a solution to
this system which balances the chemical equation. For example, a = 6, b = 6, c = 6,
and d = 1 balances the equation.

Many diverse applications are modeled by systems of equations. Systems of
equations are also important in mathematics and in particular in linear algebra. In
this chapter we develop systematic methods for solving systems of linear equations.

1.1

ß

2

Systems of Linear Equations
As the introductory example illustrates, many naturally occurring processes are
modeled using more than one equation and can require many equations in many variables. For another example, models of the economy contain thousands of equations
and thousands of variables. To develop this idea, consider the set of equations
2x − y = 2
x + 2y = 6
which is a system of two equations in the common variables x and y. A solution to
this system consists of values for x and y that simultaneously satisfy each equation.

In this example we proceed by solving the first equation for y, so that
y = 2x − 2
To find the solution, substitute y = 2x − 2 into the second equation to obtain
x + 2(2x − 2) = 6

and solving for x gives

x=2

Substituting x = 2 back into the first equation yields 2(2) − y = 2, so that y = 2.
Therefore the unique solution to the system is x = 2, y = 2. Since both of these
equations represent straight lines, a solution exists provided that the lines intersect.
These lines intersect at the unique point (2, 2), as shown in Fig. 1(a). A system of
equations is consistent if there is at least one solution to the system. If there are no
solutions, the system is inconsistent. In the case of systems of two linear equations
with two variables, there are three possibilities:
1. The two lines have different slopes and hence intersect at a unique point, as shown
in Fig. 1(a).
2. The two lines are identical (one equation is a nonzero multiple of the other), so
there are infinitely many solutions, as shown in Fig. 1(b).


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1.1 Systems of Linear Equations
x + 2y = 6


y
5

2x − y = 2

y

y

x+y =1 5

5

(2, 2)
Ϫ5

x−y =1
5

Ϫ5

(a)

−x + y = 1

x

Ϫ5

5


Ϫ5

x

2x + 2y = 2

(b)

Ϫ5

5

x

Ϫ5

(c)

Figure 1

3. The two lines are parallel (have the same slope) and do not intersect, so the
system is inconsistent, as shown in Fig. 1(c).
When we are dealing with many variables, the standard method of representing
linear equations is to affix subscripts to coefficients and variables. A linear equation
in the n variables x1 , x2 , . . . , xn is an equation of the form
a1 x1 + a2 x2 + · · · + an xn = b
To represent a system of m linear equations in n variables, two subscripts are used for
each coefficient. The first subscript indicates the equation number while the second
specifies the term of the equation.


DEFINITION 1

System of Linear Equations A system of m linear equations in n variables,
or a linear system, is a collection of equations of the form

a11 x1 + a12 x2 + · · · + a1n xn = b1





 a21 x1 + a22 x2 + · · · + a2n xn = b2
a31 x1 + a32 x2 + · · · + a3n xn = b3

..
..
..
..
..


.
.
.
.
.




am1 x1 + am2 x2 + · · · + amn xn = bm
This is also referred to as an m × n linear system.

For example, the collection of equations

 −2x1 + 3x2 + x3 − x4 = −2
x1
+ x3 − 4x4 = 1

3x1 − x2
− x4 = 3
is a linear system of three equations in four variables, or a 3 × 4 linear system.
A solution to a linear system with n variables is an ordered sequence
(s1 , s2 , . . . , sn ) such that each equation is satisfied for x1 = s1 , x2 = s2 , . . . , xn = sn .
The general solution or solution set is the set of all possible solutions.


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Chapter 1 Systems of Linear Equations and Matrices

The Elimination Method
The elimination method, also called Gaussian elimination, is an algorithm used to
solve linear systems. To describe this algorithm, we first introduce the triangular form
of a linear system.
An m × n linear system is in triangular form provided that the coefficients
aij = 0 whenever i > j. In this case we refer to the linear system as a triangular

system. Two examples of triangular systems are


 x1 − 2x2 + x3 = −1
 x1 + x2 − x3 − x4 = 2
x2 − 3x3 = 5
x2 − x3 − 2x4 = 1
and


x3 = 2
2x3 − x4 = 3
When a linear system is in triangular form, then the solution set can be obtained
using a technique called back substitution. To illustrate this technique, consider the
linear system given by

 x1 − 2x2 + x3 = −1
x2 − 3x3 = 5

x3 = 2
From the last equation we see that x3 = 2. Substituting this into the second equation,
we obtain x2 − 3(2) = 5, so x2 = 11. Finally, using these values in the first equation,
we have x1 − 2(11) + 2 = −1, so x1 = 19. The solution is also written as (19, 11, 2).

DEFINITION 2

Equivalent Linear Systems Two linear systems are equivalent if they have
the same solutions

For example, the system



 x1 − 2x2 + x3 = −1
2x1 − 3x2 − x3 = 3

x1 − 2x2 + 2x3 = 1

has the unique solution x1 = 19, x2 = 11, and x3 = 2, so the linear systems


 x1 − 2x2 + x3 = −1
 x1 − 2x2 + x3 = −1
x2 − 3x3 = 5
2x1 − 3x2 − x3 = 3
and


x3 = 2
x1 − 2x2 + 2x3 = 1
are equivalent.
The next theorem gives three operations that transform a linear system into an
equivalent system, and together they can be used to convert any linear system to an
equivalent system in triangular form.


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1.1 Systems of Linear Equations


THEOREM 1


a11 x1 + a12 x2 + · · · + a1n xn = b1




a

 21 x1 + a22 x2 + · · · + a2n xn = b2
a31 x1 + a32 x2 + · · · + a3n xn = b3

..
..
..
..
..


.
.
.
.
.



am1 x1 + am2 x2 + · · · + amn xn = bm


Let

be a linear system. Performing any one of the following operations on the linear
system produces an equivalent linear system.
1. Interchanging any two equations.
2. Multiplying any equation by a nonzero constant.
3. Adding a multiple of one equation to another.
Proof Interchanging any two equations does not change the solution of the linear
system and therefore yields an equivalent system. If equation i is multiplied by a
constant c = 0, then equation i of the new system is
cai1 x1 + cai2 x2 + · · · + cain xn = cbi
Let (s1 , s2 , . . . , sn ) be a solution to the original system. Since
ai1 s1 + ai2 s2 + · · · + ain sn = bi ,

then

cai1 s1 + cai2 s2 + · · · + cain sn = cbi

Hence (s1 , s2 , . . . , sn ) is a solution of the new linear system. Consequently, the
systems are equivalent.
For part (3) of the theorem, consider the new system obtained by adding c times
equation i to equation j of the original system. Thus, equation j of the new system
becomes
(cai1 + aj 1 )x1 + (cai2 + aj 2 )x2 + · · · + (cain + aj n )xn = cbi + bj
or equivalently,
c(ai1 x1 + ai2 x2 + · · · + ain xn ) + (aj 1 x1 + aj 2 x2 + · · · + aj n xn ) = cbi + bj
Now let (s1 , s2 , . . . , sn ) be a solution for the original system. Then
ai1 s1 + ai2 s2 + · · · + ain sn = bi

and


aj 1 s1 + aj 2 s2 + · · · + aj n sn = bj

Therefore,
c(ai1 s1 + ai2 s2 + · · · + ain sn ) + (aj 1 s1 + aj 2 s2 + · · · + aj n sn ) = cbi + bj
so that (s1 , s2 , . . . , sn ) is a solution of the modified system and the systems are
equivalent.

5


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Chapter 1 Systems of Linear Equations and Matrices

EXAMPLE 1

Use the elimination method to solve the linear system.
x+y=1
−x + y = 1

Solution

Adding the first equation to the second gives the equivalent system
x + y =1
2y = 2
From the second equation, we have y = 1. Using back substitution gives x = 0.

The graphs of both systems are shown in Fig. 2. Notice that the solution is the same
in both, but that adding the first equation to the second rotates the line −x + y = 1
about the point of intersection.

y

x+y =1 5

y

−x + y = 1

x+y =1 5

y=1

(1, 0)
Ϫ5

−x + y = 1

5

x

Ϫ5

5

x


Ϫ5

Ϫ5

(b)

(a)
Figure 2

Converting a linear system to triangular form often requires many steps. Moreover,
the operations used to convert one linear system to another are not unique and may
not be apparent on inspection. To articulate this process, the notation, for example,
(−2) · E1 + E3 −→ E3
will mean add −2 times equation 1 to equation 3, and replace equation 3 with the
result. The notation Ei ↔ Ej will be used to indicate that equation i and equation j
are interchanged.

EXAMPLE 2

Solution

Solve the linear system.


 x+y+ z= 4
−x − y + z = −2

2x − y + 2z = 2


To convert the system to an equivalent triangular system, we first eliminate the
variable x in the second and third equations to obtain